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Lines and Angles - Measuring Angles

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The protractor is a semi-circular mathematical tool used to measure angles in degrees from 0∘0^{\circ} to 180∘180^{\circ}. It has two scales: the inner scale (reading clockwise) and the outer scale (reading anti-clockwise).

A standard protractor showing degrees from 0 to 180.
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To measure an angle, place the center point of the protractor on the vertex of the angle and align the base line with one of the arms. The measure is read where the second arm crosses the scale.

Measuring a 45 degree angle with a protractor baseline alignment.
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Angles are measured in units called degrees, denoted by the symbol ∘^{\circ}. A full rotation is divided into 360360 equal parts, where each part is 1∘1^{\circ}.

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An angle can be seen as the amount of rotation of one ray around its endpoint to reach the position of another ray. For example, a 14\frac{1}{4} revolution corresponds to 90∘90^{\circ}.

📐Formulae

Right Angle=90∘\text{Right Angle} = 90^{\circ}

Straight Angle=180∘\text{Straight Angle} = 180^{\circ}

Complete Angle=360∘\text{Complete Angle} = 360^{\circ}

0∘<Acute Angle<90∘0^{\circ} < \text{Acute Angle} < 90^{\circ}

90∘<Obtuse Angle<180∘90^{\circ} < \text{Obtuse Angle} < 180^{\circ}

180∘<Reflex Angle<360∘180^{\circ} < \text{Reflex Angle} < 360^{\circ}

💡Examples

Problem 1:

Classify the following angles based on their measures: (a) 45∘45^{\circ} (b) 120∘120^{\circ} (c) 210∘210^{\circ} (d) 90∘90^{\circ}.

Solution:

(a) Acute angle, (b) Obtuse angle, (c) Reflex angle, (d) Right angle.

Explanation:

Since 45∘<90∘45^{\circ} < 90^{\circ}, it is acute. Since 90∘<120∘<180∘90^{\circ} < 120^{\circ} < 180^{\circ}, it is obtuse. Since 180∘<210∘<360∘180^{\circ} < 210^{\circ} < 360^{\circ}, it is reflex. 90∘90^{\circ} is exactly a right angle.

Problem 2:

What fraction of a revolution is a right angle?

Solution:

14\frac{1}{4}

Explanation:

A complete revolution is 360∘360^{\circ}. A right angle is 90∘90^{\circ}. The fraction is 90∘360∘=14\frac{90^{\circ}}{360^{\circ}} = \frac{1}{4}.

Problem 3:

If an angle is 12\frac{1}{2} of a straight angle, what is its measure and type?

Solution:

90∘90^{\circ}, Right Angle

Explanation:

A straight angle is 180∘180^{\circ}. Half of it is 180∘2=90∘\frac{180^{\circ}}{2} = 90^{\circ}. An angle of 90∘90^{\circ} is called a Right Angle.

Problem 4:

Identify the measure of the angle shown in the diagram and classify it as acute, obtuse, or right.

An angle of 60 degrees.

Solution:

60∘60^{\circ}, Acute Angle

Explanation:

By placing the protractor on the vertex, we see the ray passes through the 60∘60^{\circ} mark. Since 60∘<90∘60^{\circ} < 90^{\circ}, the angle is classified as an acute angle.

Problem 5:

Calculate the angle between the two rays if one ray is at the 30∘30^{\circ} mark and the other is at the 150∘150^{\circ} mark on the same scale of a protractor.

An angle formed by rays at 30 and 150 degrees.

Solution:

120∘120^{\circ}, Obtuse Angle

Explanation:

The measure of the angle between two rays is the difference between their positions on the scale: 150∘−30∘=120∘150^{\circ} - 30^{\circ} = 120^{\circ}. Since 120∘120^{\circ} is between 90∘90^{\circ} and 180∘180^{\circ}, it is an obtuse angle.